Introduction to quantum field theory
Problem 1
A particle is moving under the influence of a 2 dimensional harmonic oscillator with the potential [math]\displaystyle{ V=\tfrac{m \omega^2}{2}(X^2 + Y^2) }[/math].
- Write down the Hamiltonian in terms of the raising and lowering operators [math]\displaystyle{ \cal{H}=\cal{H} \left( \hat{a}_x,\hat{a}_x^{\dagger},\hat{a}_y,\hat{a}_y^{\dagger} \right) }[/math]. What is the explicit form of all these operators in terms of the position and momentum operators?
- Using [math]\displaystyle{ \hat{a}_x \hat{a}_y\vert \Psi_{0,0}\rangle = 0 }[/math] find the normalized wave function [math]\displaystyle{ \vert\Psi_{0,0} (p_x, p_y)\rangle }[/math]. Please pay attention to the momentum dependence of the wave function.
- Using the previous answers, calculate [math]\displaystyle{ \vert \Psi_{0,1}(p_x, p_y)\rangle }[/math], [math]\displaystyle{ \vert \Psi_{1,0}(p_x, p_y)\rangle }[/math], [math]\displaystyle{ \vert \Psi_{1,1}(p_x, p_y) \rangle }[/math], and [math]\displaystyle{ \vert \Psi_{3,1}(p_x, p_y)\rangle }[/math], where [math]\displaystyle{ \vert \Psi_{n_x,n_y}(p_x, p_y)\rangle }[/math] is the wave function in momentum space while [math]\displaystyle{ n_{x} }[/math] and [math]\displaystyle{ n_{y} }[/math] label the energy levels.
Problem 2
We solved in class the Schrödinger equation for the spectrum of the hydrogen atom as if it were isolated in the Universe. In reality, H is in an environment, which means that the Coulomb potential does not extend to infinity, but gets screened by other charges. Using first-order perturbation theory, explore the simple case of a screened Coulomb potential cut-off at a constant distance R: [math]\displaystyle{ V(r)={-\kappa\over r}\theta(R-r) }[/math], with [math]\displaystyle{ \kappa }[/math] the square of the electron charge (in Gaussian units) and [math]\displaystyle{ \theta }[/math] the Heaviside step function.
- How would you characterize the strength of the perturbation and under which conditions does perturbation theory give reliable results?
- How does the cutoff affect the spectrum? Does it lift degeneracies? Give specific answers for three values of [math]\displaystyle{ R }[/math], one for gases under standard conditions, one for liquids and one for solids (and explain how you estimate those values). Is the approximation reliable in all three cases?
- Give detailed results for the energies and the wave functions in the [math]\displaystyle{ 1s }[/math] and [math]\displaystyle{ 2s }[/math] cases.
Problem 3
Two spin-½ particles of masses [math]\displaystyle{ m_1 }[/math] and [math]\displaystyle{ m_2 }[/math], interact via a potential that may be expressed as:
[math]\displaystyle{ V = \dfrac{-e^2}{r}\left(1 + \dfrac{b}{r}\sigma^{(1)}\cdot \sigma^{(2)} \right), }[/math]
where [math]\displaystyle{ \sigma^{(i)} }[/math] corresponds to the [math]\displaystyle{ i }[/math]-th particle; [math]\displaystyle{ r = \lvert \vec{r}_{1} - \vec{r}_2 \rvert }[/math] represents the magnitude of relative separation between the two particles; and [math]\displaystyle{ b }[/math] is an arbitrary constant. Determine the bound-state energy spectrum of the combined two-particle system. What condition(s) do you need to impose upon [math]\displaystyle{ b }[/math] to avoid unphysical eigenstates?